FI 13 Curve-Based and Empirical Fixed-Income Risk Measures

Yield-based duration and convexity rest on one assumption: that a bond delivers a fixed, certain stream of cash flows. That assumption breaks down whenever a bond carries a contingency feature such as an embedded option. For a callable or putable bond, whether the option is exercised depends on where market interest rates sit relative to the coupon, so the future cash flows are not known in advance. The duration of a callable bond, for instance, does not capture the price sensitivity around the yield-to-worst, because that yield is only one of many possible outcomes.

Because bonds with embedded options do not have a well-defined yield-to-maturity, Macaulay duration and modified duration are not the right interest rate risk measures for them. The measure that fits is how far the bond’s price moves when a benchmark yield curve, most often the government par curve, is shifted. That measure is called effective duration, and it is a curve duration statistic rather than a yield duration statistic.

How an embedded call reshapes the price-yield curve

Compare a callable bond with an otherwise identical non-callable bond (matching coupon, payment schedule, maturity, and credit standing). The non-callable bond is always worth more, and that price gap equals what the embedded call option is worth, an option the issuer owns rather than the investor. When benchmark yields are high relative to the coupon, that call option is nearly worthless. When yields fall, the call becomes valuable, because the issuer can retire the debt and reborrow at the lower rates now available. The investor bears call risk: if the bond is called, the returned cash has to be put back to work at a reduced rate.

Effective duration follows from this shape. Given parallel shifts in the benchmark curve, when yields sit high (a positive curve change) both bonds show almost the same effective duration. When yields are low (a negative change in the curve), the callable bond appreciates far less than the non-callable one, because the call option limits how high the price can climb. So an embedded call reduces effective duration, most sharply when rates are falling and a call becomes likely. A smaller effective duration signals a shorter expected life, meaning a lighter weighted-average wait for the cash flows.

Effective convexity, the second-order effect of a parallel curve shift, tells the rest of the story. For the non-callable bond, the tangent line steepens as yields fall, which is positive convexity. For the callable bond, the tangent flattens as yields fall, reaches an inflection point, and beyond it the effective convexity becomes negative. At high yields, where the call is nearly worthless, both bonds show positive convexity and behave alike. As yields drop, the curves separate, and the callable bond enters the negative convexity region, where the value of the call to the issuer further caps any price gain.

How an embedded put reshapes the curve

A putable bond hands the investor the right to return it to the issuer ahead of maturity, normally at par, which protects against rising benchmark yields that would otherwise drag the price beneath par. A putable bond is therefore always worth more than an otherwise comparable non-putable bond, and that gap measures the worth of the embedded put option, which this time the investor holds. An embedded put reduces effective duration, most noticeably when rates are rising. Crucially, putable bonds always have positive effective convexity.

Effective duration and convexity matter beyond callable and putable bonds. They also apply to mortgage-backed securities (MBSs), which come from securitizing a pool of residential or commercial loans. MBS cash flows hinge on whether homeowners refinance or pay off a mortgage early, and in the United States and some other markets borrowers tend to refinance when rates fall.

The formulas

Working out effective duration closely resembles the arithmetic behind approximate modified duration.

Effective convexity parallels the approximate convexity formula.

Two differences separate these from their yield-based cousins. First, the change in the benchmark yield curve sits in the denominator, since effective duration measures risk against a parallel shift in the whole curve. Second, the prices PV₊ and PV₋ come from an option pricing model. Those models take inputs such as the length of the call protection period, the timetable of call dates alongside their call prices, an assumed credit spread over the benchmark (including any liquidity spread), a view on future interest rate volatility, and the level of market interest rates (for example, the government par curve). The analyst keeps those first four constant and then shifts the fifth up and down in parallel to obtain PV₊ and PV₋.

Effective duration and convexity can also supplement yield duration for option-free bonds, although the two do not match exactly. Whenever the model moves the government par curve, the government spot (zero) curve moves as well, though not by the same parallel amount, so the bond price does not respond exactly as it would to an equal change in yield-to-maturity. The gap between modified duration and effective duration narrows as the yield curve flattens, as time-to-maturity shortens, and as the price moves closer to par; it vanishes only in the rare case of a flat yield curve.

As curve-based statistics, effective duration and effective convexity are the right lens for judging the interest rate risk of intricate securities with uncertain cash flows, such as bonds with embedded options. Both are usually derived from prices produced by an option valuation model under specified parallel shifts of the benchmark government curve. As with the yield-based tools, together they approximate how much a bond’s full price moves, in percentage terms, for a stated curve shift.

These measures inform both issuers and investors. Take a callable bond: if the benchmark curve drops and rates slip beneath the coupon, its effective duration shrinks and its effective convexity may swing from positive to negative. For the investor that severely caps the price gain in a falling-rate environment; for the issuer it opens the door to buying the bonds back at the call price and refinancing more cheaply. For a putable bond, when the curve rises and rates climb above the coupon, effective duration and effective convexity both fall, yet convexity never leaves positive territory. That limits the price loss for the investor, who can hand the bonds back at the put price, usually par, in place of selling into a discount, and then reinvest at the higher prevailing rates.

One practical point concerns the size of the curve shift. With approximate modified duration, a smaller yield change improves accuracy. The pricing models used for callable corporate bonds and asset-backed securities (ABSs), however, build in assumptions about how issuers or borrowers behave across rate scenarios: a corporate issuer weighs its own credit spread before calling, and a mortgage borrower weighs the value of the financed home before refinancing. Because of these embedded behavioral assumptions, shrinking the benchmark change does not necessarily sharpen the estimate. Curve-based measures now sit at the center of analysis for conventional bonds and for financial liabilities alike.

Effective duration only captures how a bond reacts to a benchmark curve move if every point on the curve shifts by an equal amount. Real curves rarely move that way. Key rate duration, sometimes called partial duration, gauges how the price responds when the benchmark yield at one chosen maturity moves while the remainder of the curve stays put. It isolates how a bond responds to movements at the key maturities of the benchmark curve.

Here r_k is the kth key rate. Unlike effective duration, key rate durations reveal shaping risk: a bond’s exposure to changes in the shape of the curve, such as steepening, flattening, or twisting. The calculation mirrors that of effective duration, except that only one point on the curve is shifted at a time rather than the whole curve. Each key rate, which might be the 0.5-, 2-, 5-, 10-, 20-, or 30-year rate, is moved up and down by 1 bp; new prices PV₊ and PV₋ are generated, and Equation 4 gives the key rate duration at that maturity in isolation.

A common use is to ask how a callable bond’s price would react if short-term benchmark rates rise while longer maturities hold still, which on an upward-sloping curve amounts to a flattening. Rearranging Equation 4 gives the price change directly.

For example, the two-year Treasury note has a modified duration of 1.99. If its rate rises 25 bps with other maturities unchanged, the estimated price change is −1.99 × 0.0025 = −0.4975%.

If this portfolio is treated as an index, a manager who wants to outperform can tilt the key rate durations to match a view on how the curve will change shape, as the next example shows.

Every measure so far, from approximate modified duration to effective duration and key rate duration, is an example of analytical duration: a statistic computed from a mathematical formula. Analytical duration quietly assumes that government bond yields and credit spreads move independently, with no correlation between them. For plenty of situations that assumption tracks the price-yield relationship well enough.

In practice, fixed-income professionals also work with empirical duration, which uses historical data in statistical models that fold in the various factors that move bond prices. These estimates are built across time and across different interest rate environments, and they feed directly into portfolio decisions.

The difference between the two approaches sharpens during market stress. In a crisis, investors sell risky assets and buy government bonds. That flight to quality drives benchmark yields down while credit spreads widen, a pattern clearly visible in the first quarter of 2020 at the onset of the COVID-19 pandemic, when the 10-year US Treasury yield fell as the US corporate BB credit spread jumped.

When a government bond bears little or no credit risk, the analytical and empirical figures usually line up, since benchmark yield moves are what drive the price. For a credit-risky bond the two diverge. When benchmark yields fall in a stress episode, credit and liquidity spreads widen as expected default risk climbs. Because spreads and benchmark yields move in opposite directions in that setting, the wider spreads partly or fully cancel the fall in benchmark yields, so the bond’s price rises less than analytical duration predicts. The result is a lower empirical duration than analytical duration. Analysts have to judge how benchmark yields and credit spreads co-move before picking one approach over the other.